A Robust and Effective Smart-Card-Based Remote User Authentication Mechanism Using Hash Function

Algorithm 1

𝐸𝑋𝑃1^{𝐻𝐴𝑆𝐻}_{𝒜,𝑅𝐸𝑈𝐴𝑆}.

(1)   Eavesdrop the login request message 𝑁𝐼𝐷^∗_𝑖,𝑀₂,𝑀₃,𝑇_𝑐 during the login phase,

    where 𝑀₂=𝑀₁⊕𝑅_𝑐⊕𝑇_𝑐 =ℎ(𝐼𝐷_𝑖‖𝑋_𝑠)⊕𝑅_𝑐⊕𝑇_𝑐 and 𝑀₃=ℎ(𝐼𝐷^∗_𝑖‖𝑅_𝑐‖𝑇_𝑐).

(2)   Call 𝑅𝑒𝑣𝑒𝑎𝑙 oracle on input 𝑀₃ to retrieve the information 𝐼𝐷_𝑖, 𝑅_𝑐 and 𝑇_𝑐 as (𝐼𝐷^′_𝑖‖𝑅^′_𝑐‖𝑇^′_𝑐) ←𝑅𝑒𝑣𝑒𝑎𝑙(𝑀₃).

(3)   Using the retrieved information 𝑅^′_𝑐 and 𝑇^′_𝑐, compute 𝑢= 𝑀₂⊕𝑅^′_𝑐⊕𝑇^′_𝑐, which needs to be ℎ(𝐼𝐷_𝑖‖𝑋_𝑠).

(4)   Call 𝑅𝑒𝑣𝑒𝑎𝑙 oracle on the computed input 𝑢 to retrieve the secret number 𝑋_𝑠 of the

    server 𝑆_𝑗 as (𝐼𝐷_𝑖^′′‖𝑋^′_𝑠) ←𝑅𝑒𝑣𝑒𝑎𝑙(𝑢).

(5)   if  𝐼𝐷^′_𝑖=𝐼𝐷_𝑖^′′  then

(6)     Accept 𝑋^′_𝑠 as the correct secret number 𝑋_𝑠 of the server 𝑆_𝑗.

(7)    return 1 (Success)

(8)   else

(9)   return 0 (Failure)

(10) end if